Basic Material Properties

stressstrain

Stress ( \( \sigma \) ): Internal force per cross-sectional area

$$ \sigma = \frac{F}{A} $$

Strain ( \( \epsilon \) ): Ratio of elongation to original

$$ \epsilon = \frac{\Delta L}{L} $$

Thermal Strain ( \( \epsilon _T \) ), Thermal Stress ( \( \sigma _T \) ):

$$ \epsilon _T = \alpha \Delta L $$
$$ \sigma _T = E \alpha \Delta L $$

Where \( \alpha \) is the linear coefficient of thermal expansion.
If there are no external forces or restraints on the expansion, thermal stress is zero.

Poisson's Ratio

Ratio of lateral strain to longitudinal strain.

$$ v = - \frac{lateral}{longitudinal} $$

Hooke's Law - Young's Modulus

$$ E = \frac{\sigma}{\epsilon} $$

Theory of Bending

$$ \frac{M}{I} = \frac{\sigma}{y} = \frac{E}{R} $$

Where \( M \) is the external moment acting on the beam, \( I \) is the second moment of area, \( \sigma \) is the stress of the material, \( y \) is the distance from the neutral layer, \( E \) is Young's Modulus, and \( R \) is the radius of curvature.

\( I \) for a rectangular cross section:

$$ I = \frac{bd^3}{12} $$

Where \( b \) is parallel to the neutral layer.

\( I \) for a circular cross section:

$$ I = \frac{\pi d^4}{64} $$

Where \( d \) is the diameter of the cross section.

Parallel Axis Theorem

$$ I_{xx} = I_{gg} + Ah^2 $$

Where \( I_{gg} \) is the second moment of area passing through the centroid of body \( G \), \( I_{xx} \) is the second moment of area parallel to \( I_{gg} \) at a distance \( h \) away. \( A \) is the area of body \( G \).

Average Shear Stress ( \( \tau \) ):

$$ \tau = \frac{shearforce}{forcearea} = \frac{P}{A} $$

Where \( P \) is the internal shear force.

shear

Assuming equilibrium conditions, average shear stress on the middle body is

$$ \tau = \frac{\frac{P}{2}}{A} $$

Shear Strain

$$ \gamma \approx tan \gamma $$
$$ G = \frac{\tau}{\gamma} $$

Where \( G \) is the shear modulus.

Shear Stress in Beams

$$ \tau = \frac{VQ}{It} $$

Where \( V \) is the internal shear force, \( Q \) is the first moment of area of the half of the member's cross-section where \( t \) is measured, \( I \) is the second moment of area of the entire cross-section, \( t \) is the width of the cross-section at the point where \( \tau \) is to be determined.

Torsion

$$ \gamma = \frac{r \theta}{l} $$

Where \( r \) is the radius of the beam, \( \theta \) is the angle of distortion in radians, \( l \) is the length of the beam.

$$ \frac{ \tau }{r} = \frac{G \theta}{l} $$

The resultant torque \( T \) for a thin-walled tube:

$$ T = 2 \pi r^2 \tau t $$

Where \( t \) is the thickness of the wall.

$$ \frac{T}{J} = \frac{ \tau }{r} = \frac{G \theta}{l} $$

Where \( T \) is the resultant torque, \( J \) is the polar second moment of area, \( \tau \) is the shear stress, \( r \) is the radius of the beam, \( G \) is the shear modulus, \( \theta \) is the angle of distortion in radians, \( l \) is the length of the beam.

The polar second moment of area for a shaft of circular cross-section is

$$ J = \frac{\pi d^4}{32} $$

For a hollow shaft:

$$ J = \frac{ \pi (d_2 ^4 - d_1 ^4 )}{32} $$

Where \( d_1 \) is the inner diameter, \( d_2 \) is the outer diameter.